A quark mass definition adequate for threshold problems

Abstract

Recent calculations of heavy quark cross sections near threshold at next-to-next-to-leading order have found second-order corrections as large as first-order ones. We analyse long-distance contributions to the heavy quark potential in momentum and coordinate space and demonstrate that long-distance contributions in momentum space are suppressed as $Λ_{QCD}^2/q^2$. We then show that the long-distance sensitivity of order $Λ_{QCD} r$ introduced by the Fourier transform to coordinate space cancels to all orders in perturbation theory with long-distance contributions to the heavy quark pole mass. This leads us to define a subtraction scheme -- the `potential subtraction scheme' -- in which large corrections to the heavy quark potential and the `potential-subtracted' quark mass are absent. We compute the two-loop relation of the potential-subtracted quark mass to the $\bar{\rm MS}$ quark mass. We anticipate that threshold calculations expressed in terms of the scheme introduced here exhibit improved convergence properties.